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Take into consideration a 13\times 12 grid with A as the starting point and B as the finishing point. Each step can either be an upward or a rightward movement. Find the total number of shortest paths that lead to endpoint B .

Option: 1

520300


Option: 2

5200300


Option: 3

520030


Option: 4

52030


Answers (1)

best_answer

We must take 13 steps to the right and 12 steps up in order to go for the shortest distance to endpoint B . Let x and y each represent a step to the right and an upward step, respectively. Consequently, there are 13 x steps and 12 y steps in total.

These 13 x s and 12 ys can be arranged in any way to find the shortest path. The binomial coefficient provides the number of possible arrangements for these steps:
{ }^n \mathrm{C}_r=\frac{n !}{r !(n-r) !}
where n denotes the overall number of steps, r the number of steps taken in one particular direction, in this case to the right, and \left ( n-r \right ) the number of steps taken in the opposite direction, in this case upwards.

Using this equation, we obtain:
\begin{aligned} & { }^{25} \mathrm{C}_{12}=\frac{25 !}{12 !(25-12) !} \\ = & \frac{25 !}{12 ! \times 13 !} \\ = & 5200300 \end{aligned}
Total number of ways : 5200300 . 

Posted by

Divya Prakash Singh

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